***********************************************************************
* *
* PLIC - A LIMITED MEMORY VARIABLE METRIC ALGORITHM WITH *
* IMPROVED CONJUGACY FOR LARGE-SCALE OPTIMIZATION. *
* *
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1. Introduction:
----------------
The double-precision FORTRAN 77 basic subroutine PLIC is designed
to find a close approximation to a local minimum of a nonlinear
function F(X) with simple bounds on variables. Here X is a vector of NF
variables and F(X) is a smooth function. We suppose that NF is large
but the sparsity pattern of the Hessian matrix is not known (or the
Hessian matrix is dense). Simple bounds are assumed in the form
X(I) unbounded if IX(I) = 0,
XL(I) <= X(I) if IX(I) = 1,
X(I) <= XU(I) if IX(I) = 2,
XL(I) <= X(I) <= XU(I) if IX(I) = 3,
XL(I) = X(I) = XU(I) if IX(I) = 5,
where 1 <= I <= NF. To simplify user's work, two additional easy to use
subroutines are added. They call the basic general subroutine PLIC:
PLICU - unconstrained large-scale optimization,
PLICS - large-scale optimization with simple bounds.
All subroutines contain a description of formal parameters and
extensive comments. Furthermore, two test programs TLICU and TLICS are
included, which contain several test problems (see e.g. [2]). These
test programs serve as examples for using the subroutines, verify their
correctness and demonstrate their efficiency.
In this short guide, we describe all subroutines which can be
called from the user's program. A detailed description of the method is
given in [1]. In the description of formal parameters, we introduce a
type of the argument that specifies whether the argument must have a
value defined on entry to the subroutine (I), whether it is a value
which will be returned (O), or both (U), or whether it is an auxiliary
value (A). Besides formal parameters, we can use a COMMON /STAT/ block
containing statistical information. This block, used in each subroutine
has the following form:
COMMON /STAT/ NRES,NDEC,NIN,NIT,NFV,NFG,NFH
The arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
NRES O Positive INTEGER variable that indicates the number of
restarts.
NDEC O Positive INTEGER variable that indicates the number of
matrix decompositions.
NIN O Positive INTEGER variable that indicates the number of
inner iterations (for solving linear systems).
NIT O Positive INTEGER variable that indicates the number of
iterations.
NFV O Positive INTEGER variable that indicates the number of
function evaluations.
NFG O Positive INTEGER variable that indicates the number of
gradient evaluations.
NFH O Positive INTEGER variable that indicates the number of
Hessian evaluations.
2. Subroutines PLICU, PLICS:
----------------------------
The calling sequences are
CALL PLICU(NF,X,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
CALL PLICS(NF,X,IX,XL,XU,IPAR,RPAR,F,GMAX,IPRNT,ITERM)
The arguments have the following meaning.
Argument Type Significance
----------------------------------------------------------------------
NF I Positive INTEGER variable that specifies the number of
variables of the objective function.
X(NF) U On input, DOUBLE PRECISION vector with the initial
estimate to the solution. On output, the approximation
to the minimum.
IX(NF) I On input (significant only for PLICS) INTEGER vector
containing the simple bounds types:
IX(I)=0 - the variable X(I) is unbounded,
IX(I)=1 - the lower bound X(I) >= XL(I),
IX(I)=2 - the upper bound X(I) <= XU(I),
IX(I)=3 - the two side bound XL(I) <= X(I) <= XU(I),
IX(I)=5 - the variable X(I) is fixed (given by its
initial estimate).
XL(NF) I DOUBLE PRECISION vector with lower bounds for variables
(significant only for PLICS).
XU(NF) I DOUBLE PRECISION vector with upper bounds for variables
(significant only for PLICS).
IPAR(7) U INTEGER parameters:
IPAR(1)=MIT, IPAR(2)=MFV, IPAR(3)-unused,
IPAR(4)=IEST, IPAR(5)-unused, IPAR(6)-unused,
IPAR(7)=MF.
Parameters MIT, MFV, IEST, MF are described in Section 3
together with other parameters of the subroutine PLIC.
RPAR(9) U DOUBLE PRECISION parameters:
RPAR(1)=XMAX, RPAR(2)=TOLX, RPAR(3)=TOLF,
RPAR(4)=TOLB, RPAR(5)=TOLG, RPAR(6)=FMIN,
RPAR(7)-unused, RPAR(6)-unused, RPAR(9)-unused.
Parameters XMAX, TOLX, TOLF, TOLB, TOLG, FMIN are
described in Section 3 together with other parameters
of the subroutine PLIC.
F O DOUBLE PRECISION value of the objective function at the
solution X.
GMAX O DOUBLE PRECISION maximum absolute value of a partial
derivative of the objective function.
IPRNT I INTEGER variable that specifies PRINT:
IPRNT= 0 - print is suppressed,
IPRNT= 1 - basic print of final results,
IPRNT=-1 - extended print of final results,
IPRNT= 2 - basic print of intermediate and final
results,
IPRNT=-2 - extended print of intermediate and final
results.
ITERM O INTEGER variable that indicates the cause of termination:
ITERM= 1 - if |X - XO| was less than or equal to TOLX
in two subsequent iterations,
ITERM= 2 - if |F - FO| was less than or equal to TOLF
in two subsequent iterations,
ITERM= 3 - if F is less than or equal to TOLB,
ITERM= 4 - if GMAX is less than or equal to TOLG,
ITERM= 6 - if termination criterion was not satisfied,
but the solution is probably acceptable,
ITERM=11 - if NIT exceeded MIT,
ITERM=12 - if NFV exceeded MFV,
ITERM< 0 - if the method failed.
The subroutines PLICU, PLICS require the user supplied subroutines
OBJ and DOBJ that define the objective function and its gradient and
have the form
SUBROUTINE OBJ(NF,X,F)
SUBROUTINE DOBJ(NF,X,G)
The arguments of the user supplied subroutines have the following
meaning.
Argument Type Significance
----------------------------------------------------------------------
NF I Positive INTEGER variable that specifies the number of
variables of the objective function.
X(NF) I DOUBLE PRECISION an estimate to the solution.
F O DOUBLE PRECISION value of the objective function at the
point X.
G(NF) O DOUBLE PRECISION gradient of the objective function
at the point X.
3. Subroutine PLIC:
-------------------
This general subroutine is called from all subroutines described
in Section 2. The calling sequence is
CALL PLIC(NF,NB,X,IX,XL,XU,GF,S,XO,GO,XM,GM,XR,GR,XMAX,TOLX,TOLF,
& TOLB,TOLG,FMIN,GMAX,F,MIT,MFV,IEST,MF,IPRNT,ITERM)
The arguments NF, NB, X, IX, XL, XU, GMAX, F, IPRNT, ITERM, have the
same meaning as in Section 2. Other arguments have the following meaning:
Argument Type Significance
----------------------------------------------------------------------
GF(NF) A DOUBLE PRECISION gradient of the objective function.
S(NF) A DOUBLE PRECISION direction vector.
XO(NF) A DOUBLE PRECISION array which contains increments of
variables.
GO(NF) A DOUBLE PRECISION array which contains increments of
gradients.
XM(NF*MF) A DOUBLE PRECISION array which contains previous
increments of variables.
GM(NF*MF) A DOUBLE PRECISION array which contains previous
increments of gradients.
XR(MF) A DOUBLE PRECISION Auxiliary array.
GR(MF) A DOUBLE PRECISION Auxiliary array.
XMAX U DOUBLE PRECISION maximum stepsize; the choice XMAX=0
causes that the default value 1.0D+16 will be taken.
TOLX U DOUBLE PRECISION tolerance for the change of the
coordinate vector X; the choice TOLX=0 causes that the
default value TOLX=1.0D-16 will be taken.
TOLF U DOUBLE PRECISION tolerance for the change of function
values; the choice TOLF=0 causes that the default
value TOLF=1.0D-14 will be taken.
TOLB U DOUBLE PRECISION minimum acceptable function value;
the choice TOLB=0 causes that the default value
TOLB=FMIN+1.0D-16 will be taken.
TOLG U DOUBLE PRECISION tolerance for the Lagrangian function
gradient; the choice TOLG=0 causes that the default
value TOLG=1.0D-6 will be taken.
FMIN U DOUBLE PRECISION lower bound for the minimum function
value. It is significant only if IEST=1. If IEST=0,
the default value FMIN=-1.0D+60 will be taken.
MIT U INTEGER variable that specifies the maximum number of
iterations; the choice MIT=0 causes that the default
value 9000 will be taken.
MFV U INTEGER variable that specifies the maximum number of
function evaluations; the choice MFV=0 causes that
the default value 9000 will be taken.
IEST I INTEGER estimation of the minimum functiom value for
the line search:
IEST=0 - estimation is not used,
IEST=1 - lower bound FMIN is used as an estimation
for the minimum function value.
MF U The number of limited-memory variable metric updates
in each iteration (they use 2*MF stored vectors).
The choice MF=0 causes that the default value MF=5
will be taken.
The choice of parameter XMAX can be sensitive in many cases. First, the
objective function can be evaluated only in a relatively small region
(if it contains exponentials) so that the maximum stepsize is necessary.
Secondly, the problem can be very ill-conditioned far from the solution
point so that large steps can be unsuitable. Finally, if the problem has
more local solutions, a suitably chosen maximum stepsize can lead to
obtaining a better local solution.
The subroutine PLIC requires the user supplied subroutines OBJ
and DOBJ which are described in Section 2.
4. Verification of the subroutines:
-----------------------------------
Subroutine PLICU can be verified and tested using the program
TLICU. This program calls the subroutines TIUD14 (initiation), TFFU14
(function evaluation) and TFGU14 (gradient evaluation) containing
22 unconstrained test problems with at most 1000 variables [2]. The
results obtained by the program TLICU on a PC computer with Microsoft
Power Station Fortran compiler have the following form.
NIT= 4843 NFV= 5290 NFG= 5290 F= 0.510010981E-13 G= 0.699E-06 ITERM= 4
NIT= 351 NFV= 407 NFG= 407 F= 14.9944763 G= 0.215E-05 ITERM= 2
NIT= 112 NFV= 125 NFG= 125 F= 0.811190191E-12 G= 0.901E-06 ITERM= 4
NIT= 137 NFV= 149 NFG= 149 F= 269.499543 G= 0.119E-05 ITERM= 2
NIT= 24 NFV= 26 NFG= 26 F= 0.245413359E-11 G= 0.436E-06 ITERM= 4
NIT= 30 NFV= 31 NFG= 31 F= 0.161896358E-10 G= 0.886E-06 ITERM= 4
NIT= 41 NFV= 46 NFG= 46 F= 335.137433 G= 0.736E-06 ITERM= 4
NIT= 28 NFV= 32 NFG= 32 F= 761774.954 G= 0.494E-03 ITERM= 2
NIT= 13 NFV= 16 NFG= 16 F= 316.436141 G= 0.310E-06 ITERM= 4
NIT= 1819 NFV= 1823 NFG= 1823 F= -128.770000 G= 0.759E-05 ITERM= 2
NIT= 110 NFV= 128 NFG= 128 F= 10.7765879 G= 0.346E-05 ITERM= 2
NIT= 242 NFV= 271 NFG= 271 F= 982.273617 G= 0.262E-04 ITERM= 2
NIT= 7 NFV= 8 NFG= 8 F= 0.960843319E-14 G= 0.138E-06 ITERM= 4
NIT= 7 NFV= 9 NFG= 9 F= 0.128843969E-08 G= 0.991E-06 ITERM= 4
NIT= 2402 NFV= 2440 NFG= 2440 F= 1.92401599 G= 0.123E-05 ITERM= 2
NIT= 205 NFV= 213 NFG= 213 F= -427.404476 G= 0.114E-04 ITERM= 2
NIT= 503 NFV= 505 NFG= 505 F=-0.379921091E-01 G= 0.831E-06 ITERM= 4
NIT= 1000 NFV= 1005 NFG= 1005 F=-0.245741193E-01 G= 0.933E-06 ITERM= 4
NIT= 501 NFV= 504 NFG= 504 F= 59.5986241 G= 0.602E-10 ITERM= 4
NIT= 1971 NFV= 1975 NFG= 1975 F= -1.00013520 G= 0.928E-06 ITERM= 4
NIT= 2661 NFV= 2662 NFG= 2662 F= 2.13866377 G= 0.909E-06 ITERM= 4
NIT= 2163 NFV= 2218 NFG= 2218 F= 1.00000000 G= 0.985E-06 ITERM= 4
NITER =19170 NFVAL =19883 NGVAL =19883 NSUCC = 22
TIME= 0:00:03.58
The rows corresponding to individual test problems contain the number of
iterations NIT, the number of function evaluations NFV, the number of
gradient evaluations NFG, the final value of the objective function F,
the norm of gradient G and the cause of termination ITERM.
Subroutine PLICS can be verified and tested using the program
TLICS. This program calls the subroutines TIUD14 (initiation), TFFU14
(function evaluation), TFGU14 (gradient evaluation) containing 22 box
constrained test problems with at most 1000 variables [2]. The results
obtained by the program TLICS on a PC computer with Microsoft Power
Station Fortran compiler have the following form.
NIT= 5415 NFV= 6227 NFG= 6227 F= 0.00000000 G= 0.000E+00 ITERM= 3
NIT= 2006 NFV= 2524 NFG= 2524 F= 3926.45961 G= 0.443E-04 ITERM= 2
NIT= 92 NFV= 102 NFG= 102 F= 0.466053345E-11 G= 0.942E-06 ITERM= 4
NIT= 71 NFV= 76 NFG= 76 F= 269.522686 G= 0.546E-06 ITERM= 4
NIT= 24 NFV= 26 NFG= 26 F= 0.245413359E-11 G= 0.436E-06 ITERM= 4
NIT= 30 NFV= 31 NFG= 31 F= 0.161896358E-10 G= 0.886E-06 ITERM= 4
NIT= 39 NFV= 43 NFG= 43 F= 336.707959 G= 0.346E-06 ITERM= 4
NIT= 55 NFV= 59 NFG= 59 F= 761925.725 G= 0.103E-02 ITERM= 2
NIT= 506 NFV= 508 NFG= 508 F= 428.056916 G= 0.785E-06 ITERM= 4
NIT= 1235 NFV= 1259 NFG= 1259 F= -82.0207503 G= 0.895E-05 ITERM= 2
NIT= 17 NFV= 27 NFG= 27 F= 96517.2947 G= 0.317E-05 ITERM= 2
NIT= 56 NFV= 68 NFG= 68 F= 4994.21410 G= 0.486E-06 ITERM= 4
NIT= 7 NFV= 8 NFG= 8 F= 0.960843319E-14 G= 0.138E-06 ITERM= 4
NIT= 7 NFV= 9 NFG= 9 F= 0.128843969E-08 G= 0.991E-06 ITERM= 4
NIT= 2402 NFV= 2440 NFG= 2440 F= 1.92401599 G= 0.123E-05 ITERM= 2
NIT= 188 NFV= 189 NFG= 189 F= -427.391653 G= 0.143E-04 ITERM= 2
NIT= 503 NFV= 505 NFG= 505 F=-0.379921091E-01 G= 0.831E-06 ITERM= 4
NIT= 1000 NFV= 1005 NFG= 1005 F=-0.245741193E-01 G= 0.933E-06 ITERM= 4
NIT= 1043 NFV= 1047 NFG= 1047 F= 1654.94525 G= 0.163E-04 ITERM= 2
NIT= 2020 NFV= 2024 NFG= 2024 F= -1.00013520 G= 0.907E-06 ITERM= 4
NIT= 1613 NFV= 1614 NFG= 1614 F= 2.41354873 G= 0.998E-06 ITERM= 4
NIT= 1548 NFV= 1586 NFG= 1586 F= 1.00000000 G= 0.758E-06 ITERM= 4
NITER =19877 NFVAL =21377 NGVAL =21377 NSUCC = 22
TIME= 0:00:04.30
References:
-----------
[1] Vlcek J., Luksan L.: A conjugate direction approach to improve the
limited memory BFGS method. Research Report V-1000.
Prague, ICS AS CR, 2011.
[2] Luksan L., Vlcek J.: Sparse and partially separable test problems
for unconstrained and equality constrained optimization. Research
Report V-767, Institute of Computer Science, Academy of Sciences
of the Czech Republic, Prague, Czech Republic, 1998.